A property of a finite or infinite sequence of random variables whose joint distribution is invariant under finite permutations of indices; probabilities depend only on the multiset of observations, not their order.

Principle

Symmetry under relabeling: if permuting indices leaves joint probabilities unchanged, the sequence is exchangeable; for infinite sequences this symmetry implies a representation as a mixture of independent and identically distributed processes under suitable conditions.

Demonstration

Sampling without regard to order from an urn with unknown composition yields draws whose joint law is invariant under permutations of draw indices, so predictive probabilities depend only on counts of observed types.

Misapplication

Confusing exchangeability with independence: exchangeable variables can exhibit strong dependence despite permutation symmetry, so treating them as independent leads to incorrect inference.

Consequence

Exchangeability justifies procedures that pool information across positions and supports predictive rules based on sufficient statistics (e.g., counts); it provides a weaker structural assumption than independence that still enables coherent probabilistic modelling.

Reversal

Order-dependent processes, such as Markov chains with transition dependence on previous states, break permutation invariance and thus are the opposite concept: their joint laws change when indices are permuted.

Boundary

Definition concerns permutation invariance of joint distributions for finite or infinite sequences; it excludes weaker notions like partial exchangeability with structured block symmetries and contexts where labels carry intrinsic, non-permutable meaning (time series with causal order).

Semantic Tension

Identical distribution plus independence implies exchangeability, but exchangeability does not imply independence; the tension is between symmetry-based modelling and assumptions of factorization into independent components.

Synthesis

Exchangeability is the symmetry property that the joint distribution of a sequence of random variables is invariant under finite permutations, yielding order-agnostic probabilistic structure and enabling mixture representations and pooled predictive inference when appropriate.